Return to zggsvd.f CVS log

Up to [local] / rpl / lapack / lapack

Initial revision

    1:       SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
    2:      $                   LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
    3:      $                   RWORK, IWORK, INFO )
    4: *
    5: *  -- LAPACK driver routine (version 3.2) --
    6: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    7: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    8: *     November 2006
    9: *
   10: *     .. Scalar Arguments ..
   11:       CHARACTER          JOBQ, JOBU, JOBV
   12:       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
   13: *     ..
   14: *     .. Array Arguments ..
   15:       INTEGER            IWORK( * )
   16:       DOUBLE PRECISION   ALPHA( * ), BETA( * ), RWORK( * )
   17:       COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
   18:      $                   U( LDU, * ), V( LDV, * ), WORK( * )
   19: *     ..
   20: *
   21: *  Purpose
   22: *  =======
   23: *
   24: *  ZGGSVD computes the generalized singular value decomposition (GSVD)
   25: *  of an M-by-N complex matrix A and P-by-N complex matrix B:
   26: *
   27: *        U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
   28: *
   29: *  where U, V and Q are unitary matrices, and Z' means the conjugate
   30: *  transpose of Z.  Let K+L = the effective numerical rank of the
   31: *  matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper
   32: *  triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
   33: *  matrices and of the following structures, respectively:
   34: *
   35: *  If M-K-L >= 0,
   36: *
   37: *                      K  L
   38: *         D1 =     K ( I  0 )
   39: *                  L ( 0  C )
   40: *              M-K-L ( 0  0 )
   41: *
   42: *                    K  L
   43: *         D2 =   L ( 0  S )
   44: *              P-L ( 0  0 )
   45: *
   46: *                  N-K-L  K    L
   47: *    ( 0 R ) = K (  0   R11  R12 )
   48: *              L (  0    0   R22 )
   49: *  where
   50: *
   51: *    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   52: *    S = diag( BETA(K+1),  ... , BETA(K+L) ),
   53: *    C**2 + S**2 = I.
   54: *
   55: *    R is stored in A(1:K+L,N-K-L+1:N) on exit.
   56: *
   57: *  If M-K-L < 0,
   58: *
   59: *                    K M-K K+L-M
   60: *         D1 =   K ( I  0    0   )
   61: *              M-K ( 0  C    0   )
   62: *
   63: *                      K M-K K+L-M
   64: *         D2 =   M-K ( 0  S    0  )
   65: *              K+L-M ( 0  0    I  )
   66: *                P-L ( 0  0    0  )
   67: *
   68: *                     N-K-L  K   M-K  K+L-M
   69: *    ( 0 R ) =     K ( 0    R11  R12  R13  )
   70: *                M-K ( 0     0   R22  R23  )
   71: *              K+L-M ( 0     0    0   R33  )
   72: *
   73: *  where
   74: *
   75: *    C = diag( ALPHA(K+1), ... , ALPHA(M) ),
   76: *    S = diag( BETA(K+1),  ... , BETA(M) ),
   77: *    C**2 + S**2 = I.
   78: *
   79: *    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
   80: *    ( 0  R22 R23 )
   81: *    in B(M-K+1:L,N+M-K-L+1:N) on exit.
   82: *
   83: *  The routine computes C, S, R, and optionally the unitary
   84: *  transformation matrices U, V and Q.
   85: *
   86: *  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
   87: *  A and B implicitly gives the SVD of A*inv(B):
   88: *                       A*inv(B) = U*(D1*inv(D2))*V'.
   89: *  If ( A',B')' has orthnormal columns, then the GSVD of A and B is also
   90: *  equal to the CS decomposition of A and B. Furthermore, the GSVD can
   91: *  be used to derive the solution of the eigenvalue problem:
   92: *                       A'*A x = lambda* B'*B x.
   93: *  In some literature, the GSVD of A and B is presented in the form
   94: *                   U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
   95: *  where U and V are orthogonal and X is nonsingular, and D1 and D2 are
   96: *  ``diagonal''.  The former GSVD form can be converted to the latter
   97: *  form by taking the nonsingular matrix X as
   98: *
   99: *                        X = Q*(  I   0    )
  100: *                              (  0 inv(R) )
  101: *
  102: *  Arguments
  103: *  =========
  104: *
  105: *  JOBU    (input) CHARACTER*1
  106: *          = 'U':  Unitary matrix U is computed;
  107: *          = 'N':  U is not computed.
  108: *
  109: *  JOBV    (input) CHARACTER*1
  110: *          = 'V':  Unitary matrix V is computed;
  111: *          = 'N':  V is not computed.
  112: *
  113: *  JOBQ    (input) CHARACTER*1
  114: *          = 'Q':  Unitary matrix Q is computed;
  115: *          = 'N':  Q is not computed.
  116: *
  117: *  M       (input) INTEGER
  118: *          The number of rows of the matrix A.  M >= 0.
  119: *
  120: *  N       (input) INTEGER
  121: *          The number of columns of the matrices A and B.  N >= 0.
  122: *
  123: *  P       (input) INTEGER
  124: *          The number of rows of the matrix B.  P >= 0.
  125: *
  126: *  K       (output) INTEGER
  127: *  L       (output) INTEGER
  128: *          On exit, K and L specify the dimension of the subblocks
  129: *          described in Purpose.
  130: *          K + L = effective numerical rank of (A',B')'.
  131: *
  132: *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
  133: *          On entry, the M-by-N matrix A.
  134: *          On exit, A contains the triangular matrix R, or part of R.
  135: *          See Purpose for details.
  136: *
  137: *  LDA     (input) INTEGER
  138: *          The leading dimension of the array A. LDA >= max(1,M).
  139: *
  140: *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
  141: *          On entry, the P-by-N matrix B.
  142: *          On exit, B contains part of the triangular matrix R if
  143: *          M-K-L < 0.  See Purpose for details.
  144: *
  145: *  LDB     (input) INTEGER
  146: *          The leading dimension of the array B. LDB >= max(1,P).
  147: *
  148: *  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
  149: *  BETA    (output) DOUBLE PRECISION array, dimension (N)
  150: *          On exit, ALPHA and BETA contain the generalized singular
  151: *          value pairs of A and B;
  152: *            ALPHA(1:K) = 1,
  153: *            BETA(1:K)  = 0,
  154: *          and if M-K-L >= 0,
  155: *            ALPHA(K+1:K+L) = C,
  156: *            BETA(K+1:K+L)  = S,
  157: *          or if M-K-L < 0,
  158: *            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
  159: *            BETA(K+1:M) = S, BETA(M+1:K+L) = 1
  160: *          and
  161: *            ALPHA(K+L+1:N) = 0
  162: *            BETA(K+L+1:N)  = 0
  163: *
  164: *  U       (output) COMPLEX*16 array, dimension (LDU,M)
  165: *          If JOBU = 'U', U contains the M-by-M unitary matrix U.
  166: *          If JOBU = 'N', U is not referenced.
  167: *
  168: *  LDU     (input) INTEGER
  169: *          The leading dimension of the array U. LDU >= max(1,M) if
  170: *          JOBU = 'U'; LDU >= 1 otherwise.
  171: *
  172: *  V       (output) COMPLEX*16 array, dimension (LDV,P)
  173: *          If JOBV = 'V', V contains the P-by-P unitary matrix V.
  174: *          If JOBV = 'N', V is not referenced.
  175: *
  176: *  LDV     (input) INTEGER
  177: *          The leading dimension of the array V. LDV >= max(1,P) if
  178: *          JOBV = 'V'; LDV >= 1 otherwise.
  179: *
  180: *  Q       (output) COMPLEX*16 array, dimension (LDQ,N)
  181: *          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
  182: *          If JOBQ = 'N', Q is not referenced.
  183: *
  184: *  LDQ     (input) INTEGER
  185: *          The leading dimension of the array Q. LDQ >= max(1,N) if
  186: *          JOBQ = 'Q'; LDQ >= 1 otherwise.
  187: *
  188: *  WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
  189: *
  190: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
  191: *
  192: *  IWORK   (workspace/output) INTEGER array, dimension (N)
  193: *          On exit, IWORK stores the sorting information. More
  194: *          precisely, the following loop will sort ALPHA
  195: *             for I = K+1, min(M,K+L)
  196: *                 swap ALPHA(I) and ALPHA(IWORK(I))
  197: *             endfor
  198: *          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
  199: *
  200: *  INFO    (output) INTEGER
  201: *          = 0:  successful exit.
  202: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
  203: *          > 0:  if INFO = 1, the Jacobi-type procedure failed to
  204: *                converge.  For further details, see subroutine ZTGSJA.
  205: *
  206: *  Internal Parameters
  207: *  ===================
  208: *
  209: *  TOLA    DOUBLE PRECISION
  210: *  TOLB    DOUBLE PRECISION
  211: *          TOLA and TOLB are the thresholds to determine the effective
  212: *          rank of (A',B')'. Generally, they are set to
  213: *                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
  214: *                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
  215: *          The size of TOLA and TOLB may affect the size of backward
  216: *          errors of the decomposition.
  217: *
  218: *  Further Details
  219: *  ===============
  220: *
  221: *  2-96 Based on modifications by
  222: *     Ming Gu and Huan Ren, Computer Science Division, University of
  223: *     California at Berkeley, USA
  224: *
  225: *  =====================================================================
  226: *
  227: *     .. Local Scalars ..
  228:       LOGICAL            WANTQ, WANTU, WANTV
  229:       INTEGER            I, IBND, ISUB, J, NCYCLE
  230:       DOUBLE PRECISION   ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
  231: *     ..
  232: *     .. External Functions ..
  233:       LOGICAL            LSAME
  234:       DOUBLE PRECISION   DLAMCH, ZLANGE
  235:       EXTERNAL           LSAME, DLAMCH, ZLANGE
  236: *     ..
  237: *     .. External Subroutines ..
  238:       EXTERNAL           DCOPY, XERBLA, ZGGSVP, ZTGSJA
  239: *     ..
  240: *     .. Intrinsic Functions ..
  241:       INTRINSIC          MAX, MIN
  242: *     ..
  243: *     .. Executable Statements ..
  244: *
  245: *     Decode and test the input parameters
  246: *
  247:       WANTU = LSAME( JOBU, 'U' )
  248:       WANTV = LSAME( JOBV, 'V' )
  249:       WANTQ = LSAME( JOBQ, 'Q' )
  250: *
  251:       INFO = 0
  252:       IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
  253:          INFO = -1
  254:       ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
  255:          INFO = -2
  256:       ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
  257:          INFO = -3
  258:       ELSE IF( M.LT.0 ) THEN
  259:          INFO = -4
  260:       ELSE IF( N.LT.0 ) THEN
  261:          INFO = -5
  262:       ELSE IF( P.LT.0 ) THEN
  263:          INFO = -6
  264:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  265:          INFO = -10
  266:       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
  267:          INFO = -12
  268:       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
  269:          INFO = -16
  270:       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
  271:          INFO = -18
  272:       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
  273:          INFO = -20
  274:       END IF
  275:       IF( INFO.NE.0 ) THEN
  276:          CALL XERBLA( 'ZGGSVD', -INFO )
  277:          RETURN
  278:       END IF
  279: *
  280: *     Compute the Frobenius norm of matrices A and B
  281: *
  282:       ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
  283:       BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
  284: *
  285: *     Get machine precision and set up threshold for determining
  286: *     the effective numerical rank of the matrices A and B.
  287: *
  288:       ULP = DLAMCH( 'Precision' )
  289:       UNFL = DLAMCH( 'Safe Minimum' )
  290:       TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
  291:       TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
  292: *
  293:       CALL ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
  294:      $             TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
  295:      $             WORK, WORK( N+1 ), INFO )
  296: *
  297: *     Compute the GSVD of two upper "triangular" matrices
  298: *
  299:       CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
  300:      $             TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
  301:      $             WORK, NCYCLE, INFO )
  302: *
  303: *     Sort the singular values and store the pivot indices in IWORK
  304: *     Copy ALPHA to RWORK, then sort ALPHA in RWORK
  305: *
  306:       CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
  307:       IBND = MIN( L, M-K )
  308:       DO 20 I = 1, IBND
  309: *
  310: *        Scan for largest ALPHA(K+I)
  311: *
  312:          ISUB = I
  313:          SMAX = RWORK( K+I )
  314:          DO 10 J = I + 1, IBND
  315:             TEMP = RWORK( K+J )
  316:             IF( TEMP.GT.SMAX ) THEN
  317:                ISUB = J
  318:                SMAX = TEMP
  319:             END IF
  320:    10    CONTINUE
  321:          IF( ISUB.NE.I ) THEN
  322:             RWORK( K+ISUB ) = RWORK( K+I )
  323:             RWORK( K+I ) = SMAX
  324:             IWORK( K+I ) = K + ISUB
  325:          ELSE
  326:             IWORK( K+I ) = K + I
  327:          END IF
  328:    20 CONTINUE
  329: *
  330:       RETURN
  331: *
  332: *     End of ZGGSVD
  333: *
  334:       END